\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {1.1}The Quantum Information Revolution}{1}
\contentsline {subsection}{\numberline {1.1.1}History}{4}
\contentsline {section}{\numberline {1.2}Entanglement}{5}
\contentsline {section}{\numberline {1.3}Quantum Measurement}{5}
\contentsline {subsection}{\numberline {1.3.1}Von Neumann measurements}{5}
\contentsline {subsection}{\numberline {1.3.2}Other kinds of measurements}{5}
\contentsline {subsection}{\numberline {1.3.3}The density matrix formulation}{6}
\contentsline {subsection}{\numberline {1.3.4}Quantifying quantum information}{6}
\contentsline {subsection}{\numberline {1.3.5}Entangling measurements}{6}
\contentsline {subsection}{\numberline {1.3.6}Fidelity, purity, entropy and all that}{6}
\contentsline {section}{\numberline {1.4}Quantum state estimation}{6}
\contentsline {subsection}{\numberline {1.4.1}Techniques}{6}
\contentsline {subsection}{\numberline {1.4.2}Linear inversion}{6}
\contentsline {subsection}{\numberline {1.4.3}Maximum likelihood estimation}{6}
\contentsline {section}{\numberline {1.5}Quasi-probability distributions in quantum mechanics}{6}
\contentsline {subsection}{\numberline {1.5.1}Wigner function, Glauber-Sudarshan distributions, Husimi distributions}{6}
\contentsline {subsection}{\numberline {1.5.2}Wigner functions on a sphere}{6}
\contentsline {subsection}{\numberline {1.5.3}Discrete Wigner functions}{6}
\contentsline {subsection}{\numberline {1.5.4}Introduction to the rest of the paper}{6}
\contentsline {chapter}{\numberline {2}State Tomography}{7}
\contentsline {subsection}{\numberline {2.0.5}Properties of the density matrix}{9}
\contentsline {section}{\numberline {2.1}Measuring the density matrix}{9}
\contentsline {subsection}{\numberline {2.1.1}An example: Two-photon polarization density matrix}{12}
\contentsline {section}{\numberline {2.2}The Positivity Constraint}{12}
\contentsline {subsection}{\numberline {2.2.1}Maximum-Likelihood Estimation}{13}
\contentsline {chapter}{\numberline {3}Hidden Differences}{15}
\contentsline {section}{\numberline {3.1}Photon Modes}{16}
\contentsline {section}{\numberline {3.2}Two-mode systems}{16}
\contentsline {section}{\numberline {3.3}Visualizing multi-photon polarization states}{18}
\contentsline {subsection}{\numberline {3.3.1}Examples of Quantum Multiphoton Polarization States}{21}
\contentsline {subsection}{\numberline {3.3.2}Dicke states}{21}
\contentsline {subsection}{\numberline {3.3.3}Spin coherent states}{22}
\contentsline {subsection}{\numberline {3.3.4}NOON states}{22}
\contentsline {subsection}{\numberline {3.3.5}Characterizing two-mode states of light}{23}
\contentsline {subsection}{\numberline {3.3.6}Symmetric measurements}{26}
\contentsline {subsection}{\numberline {3.3.7}Group Theory}{27}
\contentsline {subsubsection}{The symmetric group}{27}
\contentsline {subsubsection}{Representations}{28}
\contentsline {subsubsection}{Counting the irreps}{31}
\contentsline {subsubsection}{The Schur-Weyl Duality}{33}
\contentsline {section}{\numberline {3.4}Hidden information}{40}
\contentsline {subsection}{\numberline {3.4.1}Structure of the accessible density matrix}{42}
\contentsline {subsection}{\numberline {3.4.2}Implications for Distinguishability}{44}
\contentsline {section}{\numberline {3.5}Experimental measurements}{46}
\contentsline {section}{\numberline {3.6}The two-photon experimental polarization density matrix}{48}
\contentsline {subsection}{\numberline {3.6.1}Measuring the two-photon experimental density matrix}{52}
\contentsline {subsection}{\numberline {3.6.2}Measuring the three-photon experimental density matrix}{58}
\contentsline {chapter}{\numberline {4}Quantum State Tomography with Mutually Unbiased Bases }{62}
\contentsline {section}{\numberline {4.1}Introduction}{62}
\contentsline {section}{\numberline {4.2}Comparing different tomography strategies}{64}
\contentsline {section}{\numberline {4.3}Convergence of the estimate towards the true density matrix}{66}
\contentsline {section}{\numberline {4.4}The choice of measurement bases}{68}
\contentsline {section}{\numberline {4.5}Constructing MUBs for two qubits}{71}
\contentsline {section}{\numberline {4.6}Experiment}{71}
\contentsline {subsection}{\numberline {4.6.1}Separable measurements}{72}
\contentsline {subsection}{\numberline {4.6.2}MUBs measurements}{74}
\contentsline {subsection}{\numberline {4.6.3}Analysis}{77}
\contentsline {subsection}{\numberline {4.6.4}Results}{77}
\contentsline {section}{\numberline {4.7}The Finite Wigner function for two qubits}{78}
\contentsline {section}{\numberline {4.8}Results}{78}
\contentsline {section}{\numberline {4.9}Summary}{78}
\contentsline {chapter}{\numberline {5}Measuring density matrix figures of merit without measuring the density matrix}{79}
\contentsline {section}{\numberline {5.1}Introduction}{79}
\contentsline {section}{\numberline {5.2}Multiqubit measurements}{81}
\contentsline {section}{\numberline {5.3}Experimental techniques}{86}
\contentsline {subsection}{\numberline {5.3.1}Singlet state projection}{86}
\contentsline {subsection}{\numberline {5.3.2}Making impure states}{88}
\contentsline {section}{\numberline {5.4}Results}{91}
\contentsline {subsection}{\numberline {5.4.1}Density Matrices}{91}
\contentsline {section}{\numberline {5.5}Discussion}{95}
\contentsline {chapter}{\numberline {6}Conclusions}{97}
\contentsline {section}{\numberline {6.1}Major results}{97}
\contentsline {section}{\numberline {6.2}Future outlook}{97}
